NUMERICAL LAWS AND MATHEMATICS 141 



can find numerals which, when multiplied by themselves, 

 give 2, 3, 5 . . . But a search will reveal that there are 

 no such numerals. We can find numerals which, when 

 multiplied by themselves give very nearly 2, 3, 5 . . . ; 

 for instance, 1-41, 173, 2-24 give 1-9881, 2*9929, 5-0166, 

 and we could find numerals which would come even 

 closer to those desired. And that is really all we want, 

 for our measurements are never perfectly accurate, and 

 if we can get numerals which agree very nearly with 

 our rule, that is all that we can expect. But the search 

 for such numerals would be a very long and tedious 

 business ; it would involve our drawing up an enormously 

 complicated multiplication table, including not only 

 whole numbers but also fractions with many decimal 

 places. And so the question arises if we cannot find 

 some simpler rule for obtaining quickly the number 

 which multiplied by itself will come as close as we please 

 to 2, 3, 4 ... Well, we can ; the rule is given in every 

 textbook of arithmetic ; it need not be given here. 

 The point which interests us is that, just as the simple 

 multiplication of two numerals suggested a new process, 

 namely the multiplication of a numeral by itself, so this 

 new process suggests in its turn many other and more 

 complicated processes. To each of these new processes 

 corresponds a new rule for relating numerals and for 

 arriving at one starting from another ; and to each new 

 rule may correspond a numerical law. We thus get many 

 fresh forms of numerical law suggested, and some of 

 them will be found to represent actual experiments. 



This process for extending arithmetical operations 

 beyond the simple division and multiplication from 

 which we start ; the consequent invention of new rules 

 for relating numerals and deriving one from another ; 

 and the study of the rules, when they are invented all 

 this is a purely intellectual process. It does not depend 

 on experiment at all ; experiment enters only when we 



